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The Maxima on-line user's manual

Algebra Calculator

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Set_partitions Calculator

Set_partitions

Function: set_partitions (<a>)

makelist (belln (i), i, 0, 6);
 is (cardinality (set_partitions ({})) = belln (0));
 is (cardinality (set_partitions ({1, 2, 3, 4, 5, 6})) =                                 belln (6));
 [belln (x), belln (sqrt(3)), belln (-9)];

Function: set_partitions (<a>, <n>) Returns the set of all partitions of <a>, or a subset of that set.

set_partitions(<a>, <n>) returns a set of all decompositions of <a> into <n> nonempty disjoint subsets.

set_partitions(<a>) returns the set of all partitions.

stirling2 returns the cardinality of the set of partitions of a set.

A set of sets P is a partition of a set S when

1. each member of P is a nonempty set,

2. distinct members of P are disjoint,

3. the union of the members of P equals S.

Examples:

The empty set is a partition of itself, the conditions 1 and 2 being vacuously true.

          (%i1) set_partitions ({});
          (%o1)                         {{}}

The cardinality of the set of partitions of a set can be found using stirling2.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) cardinality(p) = stirling2 (6, 3);
          (%o3)                        90 = 90

Each member of p should have <n> = 3 members; lets check.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) map (cardinality, p);
          (%o3)                          {3}

Finally, for each member of p, the union of its members should equal s; again lets check.

          (%i1) s: {0, 1, 2, 3, 4, 5}$
          (%i2) p: set_partitions (s, 3)$
          (%i3) map (lambda ([x], apply (union, listify (x))), p);
          (%o3)                 {{0, 1, 2, 3, 4, 5}}

(%o1)                                true
(%i2) 

Set_partitions Example

Related Examples

set_partitions

ts: {a, b, c, d};

set_partitions(ts,2);

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set_partitions

set_partitions({1,2,3});

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set_partitions

? set_partitions;

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set_partitions

ts: {a, b, c, d};

set_partitions(ts,2);

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set_partitions

set_partitions({1,2,3});

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set_partitions

? set_partitions;

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